arXiv subjects

Erbol Zhanpeisov

Publications and source records attributed to Erbol Zhanpeisov.

6 recordsLinked to original sources

arXiv2026

Infinite time blow-up and slow decay for the six dimensional energy-critical heat equation with self-similarly decaying initial data

We consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions that decay strictly more slowly than the self-similar rate. Moreover, the blow-up and decay rates are not uniquely determined by the decay rate of the initial data, but exhibit a certain flexibility depending on the construction. The proof is based on gluing suitably rescaled bubbles to forward self-similar solutions.

math.AP↗

arXiv2025

Blow-up rate for the subcritical semilinear heat equation in non-convex domains

We consider the semilinear heat equation $u_t=Δu+|u|^{p-1} u$ in possibly non-convex and unbounded domains. Our main result shows the nonexistence of type II blow-up for possibly sign-changing solutions in the energy subcritical range $(n-2)p<n+2$. This resolves a long-standing open question dating back to the 1980s and also deduces the blow-up of the scaling critical norm.

math.AP↗

arXiv2025

Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy

We study a class of fully nonlinear boundary-degenerate elliptic equations, for which we prove that u \equiv 0 is the only solution. Although no boundary conditions are posed together with the equations, we show that the operator degeneracy actually generates an implicit boundary condition. Under appropriate assumptions on the degeneracy rate and regularity of the operator, we then prove that there exist no bounded solutions other than the trivial one. Our method is based on the arguments for uniqueness of viscosity solutions to state constraint problems for Hamilton-Jacobi equations. We obtain similar results for fully nonlinear degenerate parabolic equations. Several concrete examples of equations that satisfy the assumptions are also given.

math.AP↗

arXiv2023

Existence of solutions for time fractional semilinear parabolic equations in Besov--Morrey spaces

We consider the Cauchy problem for a time fractional semilinear heat equation with initial data belonging to inhomogeneous/homogeneous Besov--Morrey spaces. We present sufficient conditions for the existence of local/global-in-time solutions to the Cauchy problem, which cover all existing results in the literature and can be applied to a large class of initial data.

math.AP↗

arXiv2023

Existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces

In this paper, we establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.

math.AP↗

arXiv2020

Blow-up rate of sign-changing solutions to nonlinear parabolic systems

We present a blow-up rate estimate for a solution to the parabolic Gross-Pitaevskii and related systems on entire space with Sobolev subcritical nonlinearity. We extend the results of [Y. Giga, S. Matsui and S. Sasayama, Indiana Univ. Math. J. {53} (2004), 483--514] to the parabolic systems.

math.AP↗